Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classification of tight contact structures on some Seifert fibered manifolds: I

Tanushree Shah

TL;DR

The article classifies tight contact structures with zero Giroux torsion on toroidal Seifert-fibered manifolds with four exceptional fibers, providing an exact count in the main torus-bounded family $M(0,e_0;p_1/q_1, \dots,p_4/q_4)$ for $e_0\le -4$. The authors derive a sharp lower bound via Legendrian surgery (creating Stein-fillable structures) and obtain a matching upper bound through convex surface theory and a detailed analysis of decomposed blocks, ultimately proving $|\,\pi_0(Tight^{min}(M))\,|=|e_0(M)+1|\prod_{i=1}^4\prod_{j=1}^{m_i}(a_j^i+1)$. They illustrate the method with a concrete example $M(0,-4;1/2,1/2,1/2,1/2)$, where exactly three such structures exist, and extend the result to the general family, while discussing the presence of $ obreak\mathbb{Z}$-many non-isotopic tight structures with nonzero torsion on toroidal bases. The work relies on Legendrian surgery, LM-type Chern-class distinctions, and Honda–Eliashberg convex-surface classifications to combine local block data into global counts, with all zero-torsion cases shown to be Stein fillable.

Abstract

We classify tight contact structures with zero Giroux torsion on some Seifert-fibered manifolds with four exceptional fibers. We get the lower bound by constructing contact structures using Legendrian surgery. We use convex surface theory to obtain the upper bound.

Classification of tight contact structures on some Seifert fibered manifolds: I

TL;DR

The article classifies tight contact structures with zero Giroux torsion on toroidal Seifert-fibered manifolds with four exceptional fibers, providing an exact count in the main torus-bounded family for . The authors derive a sharp lower bound via Legendrian surgery (creating Stein-fillable structures) and obtain a matching upper bound through convex surface theory and a detailed analysis of decomposed blocks, ultimately proving . They illustrate the method with a concrete example , where exactly three such structures exist, and extend the result to the general family, while discussing the presence of -many non-isotopic tight structures with nonzero torsion on toroidal bases. The work relies on Legendrian surgery, LM-type Chern-class distinctions, and Honda–Eliashberg convex-surface classifications to combine local block data into global counts, with all zero-torsion cases shown to be Stein fillable.

Abstract

We classify tight contact structures with zero Giroux torsion on some Seifert-fibered manifolds with four exceptional fibers. We get the lower bound by constructing contact structures using Legendrian surgery. We use convex surface theory to obtain the upper bound.
Paper Structure (12 sections, 13 theorems, 3 equations, 24 figures)

This paper contains 12 sections, 13 theorems, 3 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Giroux torsion
  4. Tight structures on Seifert manifolds
  5. Example case of Theorem \ref{['general thm']}
  6. Upper bound
  7. Maximising twisting numbers
  8. Combining tight contact structures on basic blocks
  9. General case
  10. Upper Bound
  11. Combining tight contact structures on the basic blocks
  12. Concluding remarks

Key Result

Theorem 1

Let $M=M(e_0,0;p_1/q_1,p_2/q_2, p_3/q_3,p_4/q_4)$ where $e_0\leq -4$ and $\frac{p_i}{q_i}\in (0,1)\cap \mathbb{Q}$ and $gcd(p_i,q_i)=1$. On $M$ there are exactly $|(e_0(M)+1)\Pi_{i=1}^4\Pi_{j=1}^{m_i}(a_j^i+1)|$ tight contact structures with zero Giroux torsion up to contact isotopy. All of these ca

Figures (24)

  • Figure 1: Surgery description of the Seifert manifold.
  • Figure 2: Legendrian realization of unknot with Thurston-Bennequin number -1.
  • Figure 3: Legendrian realizations of the unknot with Thurston-Bennequin number -3.
  • Figure 4: Decomposition of a Seifert fibered 3-manifold with four exceptional fibers.
  • Figure 5: Dividing curve on $\Sigma$ with contact structure $\xi_A$ on $\Sigma\times S^1$.
  • ...and 19 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem : Special case of Theorem \ref{['general thm']}
  • ...and 8 more